Case b > 0 : I ( N ) is a decreasing convex function.
If is a decreasing convex function on and, then the reverse inequality in (3.4) holds.
If φ is a concave function on I and ψ is a decreasing convex function on φ ( I ), then ψ ∘ φ is convex on I. Proof Straightforward.
The intensity mapping function (5) is a decreasing convex function with p < 1 and a decreasing concave function with p > 1 as shown in Figure 3.
(a) Let G : [ 0, ∞ ) → [ 0, ∞ ) be a decreasing and log-convex function, with G ( x ) > 0, x ≥ 0 and right-continuous at x = 0. Then E G [ X ( t ) ] is a log-convex function on [ 0, ∞ ).
Let G : ( 0, ∞ ) → [ 0, ∞ ) be a decreasing and log-convex function, with G ( x ) > 0, x > 0. Assume that P X X ( t ) = 0 ) = 0 and that E G [ X ( t ) ] < ∞, for all t > 0. Then E G [ X ( t ) ] is a log-convex function on ( 0, ∞ ).
Let G : [ 0, ∞ ) → [ 0, ∞ ) be a decreasing and log-convex function, with G ( x ) > 0, x ≥ 0 and right-continuous at x = 0. Then E G [ X ( t ) ] is a log-convex function on [ 0, ∞ ).
Let P be a normal cone, A : P → P be an increasing α 1 -concave operator and B : P → P be a decreasing α 2 -convex operator.
The yield differential between notes and bills is a decreasing and convex function of the time to maturity.
Let (f:[0,infty tomathbb{R}) be a function satisfying the following conditions: (f1): f is a continuous function, (fgeq0); (f2): f is a decreasing and convex function; (f3): (f(0)=1), (0< f (frac{M}{8} )<1); (f gammaf(gamma x geqfrac{1}{2}) for every (xin [0,frac {M}{8} ]), where gamma=1-frac{1}{6}f biggl(frac{M}{8} biggr) biggl 1-f biggl 1-fc {m}{2} biggl) biggr).
